1.01c Disproof by counter example

State whether the following statements are true or false; if false, provide a counter-example.

1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational. [3]

Prove that the following statement is false. For all integers \(n\) greater than or equal to 1, \(n^2 + 3n + 1\) is a prime number. [2]

A student was attempting to prove that \(x = \frac{1}{2}\) is the only real root of $$x^3 + \frac{1}{4}x - \frac{1}{2} = 0.$$ The attempted solution was as follows. $$x^3 + \frac{1}{4}x = \frac{1}{2}$$ $$\therefore \quad x(x^2 + \frac{1}{4}) = \frac{1}{2}$$ $$\therefore \quad x = \frac{1}{2}$$ or $$x^2 + \frac{1}{4} = \frac{1}{2}$$ i.e. $$x^2 = -\frac{1}{4} \quad \text{no solution}$$ $$\therefore \quad \text{only real root is } x = \frac{1}{2}$$

1. Explain clearly the error in the above attempt. [2]
2. Give a correct proof that \(x = \frac{1}{2}\) is the only real root of \(x^3 + \frac{1}{4}x - \frac{1}{2} = 0\). [3]

The equation $$x^3 + \beta x - \alpha = 0 \quad \text{(I)}$$ where \(\alpha\), \(\beta\) are real, \(\alpha \neq 0\), has a real root at \(x = \alpha\).

1. Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(|\alpha| < 2\). [6]

An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation (I) but the incorrect method used by the student produces 3 distinct real "roots".

1. Find the range of possible values for \(\alpha\). [7]

Dan believes that for every positive integer \(n\), at least one of \(2^n - 1\) and \(2^n + 1\) is prime. Which value of \(n\) shown below is a counter example to Dan's belief? Circle your answer. [1 mark] \(n = 3\) \(n = 4\) \(n = 5\) \(n = 6\)

A student suggests that for any positive integer \(n\) the value of the expression $$4n^2 + 3$$ is always a prime number. Prove that the student's statement is false by finding a counter example. Fully justify your answer. [3 marks]

Use a counter example to show that the following statement is false. "\(n^2 - n + 5\) is a prime number, for \(2 \leq n \leq 6\)"

Tom conjectures that if \(n\) is an odd number greater than 1, then \(2^n - 1\) is prime. Find a counter example to disprove Tom's conjecture. [3]

A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"

1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
2. Refine the student's claim to make it fully correct. [1 mark]
3. Prove that your answer to part (b) is correct. [3 marks]

In each of the two statements below, \(x\) and \(y\) are real numbers. One of the statements is true while the other is false. A: \(x^2 + y^2 \geqslant 2xy\), for all real values of \(x\) and \(y\). B: \(x + y \geqslant 2\sqrt{xy}\), for all real values of \(x\) and \(y\).

1. Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]

Show, by counter example, that the following statement is false. "For all positive integer values of \(n\), \(n^2 + 1\) is a prime number." [3]

In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true while the other is false. A: Given that \((2c + 1)^2 = (2d + 1)^2\), then \(c = d\). B: Given that \((2c + 1)^3 = (2d + 1)^3\), then \(c = d\).

1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [5]

Prove or disprove each of the following statements:

1. If \(n\) is an integer, then \(3n^2 - 11n + 13\) is a prime number. [2]
2. If \(x\) is a real number, then \(x^2 - 8x + 17\) is positive. [2]
3. If \(p\) and \(q\) are irrational numbers, then \(pq\) is irrational. [2]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find